Lower semi-continuity of the Pareto solution map in quasiconvex semi-infinite vector optimization

AbstractThis paper is concerned with the lower semi-continuity of the efficient (Pareto) solution map for the perturbed quasiconvex semi-infinite vector optimization problem (QCSVO). We establish sufficient conditions for the lower semi-continuous property of the efficient solution map of (QCSVO) under functional perturbations of both the objective function and the constraints. Examples are designed to analyze the obtained results

Calmness of efficient solution maps in parametric vector optimization

The paper is concerned with the stability theory of the efficient solution map of a parametric vector optimization problem. Utilizing the advanced tools of modern variational analysis and generalized differentiation, we study the calmness of the efficient solution map. More explicitly, new sufficient conditions in terms of the Fréchet and limiting coderivatives of parametric multifunctions for this efficient solution map to have the calmness at a given point in its graph are established by employing the approach of implicit multifunctions. Examples are also provided for analyzing and illustrating the results obtained. © 2011 Springer Science+Business Media, LLC

Conic Relaxations with Stable Exactness Conditions for Parametric Robust Convex Polynomial Problems

In this paper, we examine stable exact relaxations for classes of parametric robust convex polynomial optimization problems under affinely parameterized data uncertainty in the constraints. We first show that a parametric robust convex polynomial problem with convex compact uncertainty sets enjoys stable exact conic relaxations under the validation of a characteristic cone constraint qualification. We then show that such stable exact conic relaxations become stable exact semidefinite programming relaxations for a parametric robust SOS-convex polynomial problem, where the uncertainty sets are assumed to be bounded spectrahedra. In addition, under the corresponding constraint qualification, we derive stable exact second-order cone programming relaxations for some classes of parametric robust convex quadratic programs under ellipsoidal uncertainty sets.Chuong was supported by the National Foundation for Science and Technology Development of Vietnam (NAFOSTED) under grant number 101.01−2020.09. Research of J. Vicente-Pérez was partially supported by the Ministry of Science, Innovation and Universities of Spain and the European Regional Development Fund (ERDF) of the European Commission, Grant PGC2018-097960-B-C22, and by the Generalitat Valenciana, Grant AICO/2021/165